3.919 \(\int \frac{1}{x^5 \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{\sqrt{x^4+1}}{4 x^4} \]

[Out]

-Sqrt[1 + x^4]/(4*x^4) + ArcTanh[Sqrt[1 + x^4]]/4

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Rubi [A]  time = 0.0358663, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{\sqrt{x^4+1}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(4*x^4) + ArcTanh[Sqrt[1 + x^4]]/4

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Rubi in Sympy [A]  time = 3.95533, size = 24, normalized size = 0.77 \[ \frac{\operatorname{atanh}{\left (\sqrt{x^{4} + 1} \right )}}{4} - \frac{\sqrt{x^{4} + 1}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**4+1)**(1/2),x)

[Out]

atanh(sqrt(x**4 + 1))/4 - sqrt(x**4 + 1)/(4*x**4)

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Mathematica [A]  time = 0.0384936, size = 31, normalized size = 1. \[ \frac{1}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{\sqrt{x^4+1}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(4*x^4) + ArcTanh[Sqrt[1 + x^4]]/4

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Maple [A]  time = 0.013, size = 24, normalized size = 0.8 \[ -{\frac{1}{4\,{x}^{4}}\sqrt{{x}^{4}+1}}+{\frac{1}{4}{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{4}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^4+1)^(1/2),x)

[Out]

-1/4*(x^4+1)^(1/2)/x^4+1/4*arctanh(1/(x^4+1)^(1/2))

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Maxima [A]  time = 1.43835, size = 50, normalized size = 1.61 \[ -\frac{\sqrt{x^{4} + 1}}{4 \, x^{4}} + \frac{1}{8} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) - \frac{1}{8} \, \log \left (\sqrt{x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^5),x, algorithm="maxima")

[Out]

-1/4*sqrt(x^4 + 1)/x^4 + 1/8*log(sqrt(x^4 + 1) + 1) - 1/8*log(sqrt(x^4 + 1) - 1)

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Fricas [A]  time = 0.260686, size = 59, normalized size = 1.9 \[ \frac{x^{4} \log \left (\sqrt{x^{4} + 1} + 1\right ) - x^{4} \log \left (\sqrt{x^{4} + 1} - 1\right ) - 2 \, \sqrt{x^{4} + 1}}{8 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^5),x, algorithm="fricas")

[Out]

1/8*(x^4*log(sqrt(x^4 + 1) + 1) - x^4*log(sqrt(x^4 + 1) - 1) - 2*sqrt(x^4 + 1))/
x^4

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Sympy [A]  time = 6.29972, size = 22, normalized size = 0.71 \[ \frac{\operatorname{asinh}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**4+1)**(1/2),x)

[Out]

asinh(x**(-2))/4 - sqrt(1 + x**(-4))/(4*x**2)

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GIAC/XCAS [A]  time = 0.227321, size = 50, normalized size = 1.61 \[ -\frac{\sqrt{x^{4} + 1}}{4 \, x^{4}} + \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{4} + 1} + 1\right ) - \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^5),x, algorithm="giac")

[Out]

-1/4*sqrt(x^4 + 1)/x^4 + 1/8*ln(sqrt(x^4 + 1) + 1) - 1/8*ln(sqrt(x^4 + 1) - 1)